Directions. Please submit your assignment typed. If there is more than one page, please staple.
Explain your solutions.
Encrypt the string ``Sell everything at once!'' one character at a time by forming the sequence consisting of the 7^{th} powers of its decimal ASCII codes modulo 11021.
The method of the previous problem was used to encrypt a string. The corresponding encrypted sequence is
| [5965, 5730, 8607, 5425, 711, 7508, 711, 1237, 6720, 5730, 7508, 327, |
| 7508, 5730, 1237, 9857, 8738, 711, 7508, 5730, 8215, 8607, 711, 8215, |
| 8607, 1332, 8215, 9821, 3953] |
Let m be the number
| 5258168548013597450620892673749432457952803197 . |
Repeat the previous exercise with the following modification. Instead of operating on the decimal ASCII code of each of the nine characters in the string, take the characters 3 at a time, and form the length 3 sequence consisting of the numbers c + b . 128 + a . (128)^{2} for each of the three sequences of numbers [a, b, c] appearing in the sequence of 9 ASCII codes of the string. (Notice that this technique is both more efficient and more ``secure'' than that of the previous exercise.) For ease of transcription you may respond to this exercise by submitting the least non-negative residues mod 128 of the encrypted sequence.
What step in trying to decrypt a string encoded with the known exponent 10001 modulo the number m by the method of the previous exercise makes such decryption difficult? Are you able to find the 3 character string encoded that way to produce the single term sequence
| [ 394070967091950488701702235969152190968955616 ] ? |